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Recent contributions to Implementation/Mechanism Design Theory

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Recent contributions toImplementation/Mechanism Design Theory

E. Maskin

- part of economic theory devoted to “reverse engineering”
- usually we take mechanism, game, or economy as given
- try to predict the outcomes it generates in equilibrium

- in MD, we (the “planner”) start with outcome(s) a we want as a function of underlying state
technologies, endowments, etc.)

social choice function

- difficulty: we may not know state
- try to design a mechanism (game, outcome function, tax schedule) whose equilibrium outcomes same as that prescribed by social choice function
mechanism implements SCF

- Goes back (at least) to 19th century Utopians
- can one design “humane” alternative to laissez-faire capitalism?

- Socialist Planning Controversy 1920s-40s
- can one construct a centralized planning mechanism that replicates or improves on competitive markets?
O. Lange and A. Lerner: yes

L. von Mises and F. von Hayek: no

- brought to fore 2 major themes
incentives

information

- can one construct a centralized planning mechanism that replicates or improves on competitive markets?

Formal mechanism-design theory dates from

3 papers

- L. Hurwicz (1960)
- introduced basic concepts
- mechanism
- informational decentralization
- informational efficiency

- introduced basic concepts
- W. Vickrey (1961)
- exhibited a particular but important mechanism: 2nd price auction

- J. Mirrlees (1971)
- developed standard analytic techniques
- derived standard properties (e.g., “no distortion at top”)

Since then, field has expanded dramatically

- vast literature, ranging from
- very general
possible outcomes abstract set of social alternatives

(at least 10 major survey articles and books in last dozen years or so)

- quite particular
design of bilateral contracts between buyer and seller

(several recent books on contract theory, including Bolton-Dewatripont (2005) and Laffont-Martimort (2002))

design of auctions for allocating a good among competing bidders

(several recent books - - Krishna (2002), Milgrom (2004), Klemperer (2004))

- far too much recent work to survey properly here
- will pick 3 specific developments (both general and particular)

- very general

- interdependent values in auction design
- robustness of mechanisms
- indescribable states, renegotiation and incomplete contracts

- seller has 1 good
- n potential buyers
- how to allocate good efficiently?
(to buyer who values good the most)

i.e., how to implement SCF that selects efficient allocation

In private values case (each buyer’s valuation is independent of others’ information),

Vickrey (1961) answered question:

- 2nd price auction is efficient
- buyers submit bids
- winner is high bidder
- winner pays 2nd highest bid

- if is buyer i’s valuation, optimal for him to bid
- winner will have highest valuation

- each buyer i gets private signal (one-dimensional)
- buyer i’s valuation is
- buyer i no longer knows own valuation
- so can’t bid valuation in equilibrium
- might bid expected valuation, but this not enough for efficiency: might have

- consider auction in which
- each buyer i makes contingent bid
- calculate fixed point such that
- winner is buyer i such that
- winner pays

- If (i)
then, in equilibrium buyer i with signal bids true contingent valuation:

- idea: i pays lowestconstant bid that would win
- in ordinary 2nd price auction, i pays lowest bid

- from true valuations are fixed point
- from , winner has highest valuation
- auction efficient

- dynamic auctions like English auction easier on buyers than one-shot mechanisms like 2nd –price auction
- Perry and Reny (2005) develop dynamic auction
- even works for multiple goods, if substitutes

- open problem: How to handle multiple goods with complementarities in dynamic auction
- Jehiel and Moldovanu (2002) and Jehiel, Moldovanu, Meyer-Ter-Vehn, and Zame (2005) establish fundamental limitation on how far one can go with multiple goods and multidimensional signals

Robust Mechanism Design

auction in which buyer i bids is “robust” or “independent of detail” in sense that

- it doesn’t matter whether auction designer knows buyers’ signal spaces or functional forms
- it doesn’t matter what buyer i believes about the distribution of
- optimal for buyer i to set
regardless of i’s belief about

- i.e., bidding truthfully is an ex post equilibrium
(remains equilibrium even if iknows )

- optimal for buyer i to set

Why is robustness important?

- common in Bayesian mechanism design to identify buyer i’s possible types with his possible preferences (common more generally than justified)
set of possible types set of possible preferences

- but this has extreme implication: if you know i’s preferences, know his beliefs over other’s types
- no reason why this should hold
- very strong consequences:
in auction model above, if signals correlated, auctioneer can attain

efficiency and extract all buyer surplus, even without conditions such as

(Crémer and McLean (1985))

- As Neeman (2001) and Heifetz and Neeman (2004) show, Crémer-McLean result goes away for suitably rich type spaces (preference corresponds to multiple possible beliefs)

- no reason why auction designer should know what buyers’ type spaces are (how preferences related to beliefs)

Given SCF , can we find mechanism for which, regardless of type space associated with preference space , there always exists f-optimal equilibrium?

(robust partial implementation)

- sufficient condition: f partially implementable in ex post equilibrium, i.e., there exists mechanism that always has f-optimal ex post equilibrium (may be other equilibria)
- ex post equilibrium reduces to dominant strategy equilibrium with private values

- interestingly, Bergemann and Morris (2004) show that condition not necessary

But ex post partial implementability is necessary for robust partial implementation if

- outcome space takes form
agent i cares just about

- satisfied in above auction model (and, more generally, in quasilinear models)

- So far have concentrated on partial implementation (not all equilibria have to be f-optimal)
- But unless planner sure that agents will play f-optimal equilibrium, more appropriate concept is full implementation: all equilibria of mechanism must be f-optimal

- key to full implementation is some species of monotonicity
- full implementation in Nash equilibrium (agents have complete information) requires standard monotonicity:
for all,

then

equivalently: for all for which

there exist i and such that

analogous condition for Bayesian implementation (agents have incomplete information)

- - Postlewaite and Schmeidler (1986), Palfrey and Srivastava (1989), Jackson (1991)

- full implementation in Nash equilibrium (agents have complete information) requires standard monotonicity:

- standard monotonicity: for all
there exist

- ex post monotonicity key to ex post full implementabilty (Bergemann and Morris 2005):
for all and

there exist i and a such that

and

- in economic settings with n > 3,f is expost fully implementable if and only if it satisfies ex post monotonicity and ex post incentive compatibility
- ex post equilibrium is refinement of Nash equilibrium but ex post monotonicity doesn’t imply standard monotonicity (nor is it implied)
- although ex post equilibrium is more demanding solution concept, makes ruling out equilibria easier

- Notable SCF where ex post monotonicity but not standard monotonicity satisfied: efficient generalized second-price allocation rule in interdependent values auction model when n> 3
- ex post monotonicity satisfied because truthful equilibrium is unique ex post equilibrium
- Berliun (2003) shows that hypothesis n> 3 is important: there exist inefficient ex post equilibria in case n = 2.

- efficient second-price allocation rule not standardly monotonic: if lower losers’ signal values, monotonicity requires that same allocation still chosen - - but winner’s payment will fall

- But ex post full implementation not quite enough
- ensures ex post equilibria optimal
- but other equilibria could be nonoptimal

- really need robust full implementation:
implementable by mechanism such that, regardless of type space associated with Θ, all equilibria are f-optimal

- robust monotonicity is key:
for all

and

for all there exists

such that

and

- stronger than both ex post monotonicity and standard monotonicity
- together with ex post incentive compatibility, necessary and sufficient for robust full implementation in economic environments

- For n> 3, generalized 2nd price allocation rule robustly fully virtually implementable as long as not “too much” interdependence, i.e.,

- so far, “robustness” requirement pertains to mechanism designer
- may not know agents’ type spaces

- also recent contributions in which robustness pertains to agents playing mechanism

- large literature considering implementation in various refinements of Nash equilibrium
- allows implementation of nonmonotonic SCFs

- any species of Nash equilibrium entails that agents have common knowledge of preferences, i.e.,
state

- but what if agents are (slightly ) uncertain about
?

- which SCFs are robust to this uncertainty?
- answer depends on nature of uncertainty

- for a natural form of uncertainty, only monotonic SCFs robustly implementable in this sense (Chung and Ely 2003)

Example (Jackson and Srivastava (1996))

- If , so mechanism no longer implements f
- in fact, no mechanism can implement f because nonmonotonic

Open problems:

(1) Implications of ex post monotonicity and robust monotonicity for applications

(2) implications of other sorts of uncertainty for implementability

Indescribable States, Renegotiation and Incomplete Contracts

- incomplete contracts literature studies how assigning ownership (or control) of productive assets affects efficiency of outcome
- For efficiency to be in doubt, must be some constraint on contracting
(i.e., on mechanism design)

- In this literature, constraint is incompleteness of contract
- contract not as fully contingent on state of world as parties would like

- Reason for incompleteness
- parties plan to trade some good in future
- do not know characteristics of good (state) at the time of contracting (although common knowledge at time of trade)
- contract cannot even describe set of possible states (too vast)
- so contract cannot be fully contingent

Nevertheless, we have:

Irrelevance Theorem:

If parties are risk averse and can assign probability distribution to their future payoffs, then can achieve same expected payoffs as with fully contingent contract (even though cannot describe possible states in advance)

Idea:

- design contracts that specify payoff contingencies
- later, when state of world realized, can fill in physical details
- possible problem: incentive compatibility
- will it be in parties interest to specify physical details truthfully?
- but if
can design mechanism to ensure incentive compatibility

- make mechanism part of contract

Where does risk aversion come in?

Answer: helps with incentive compatibility

- if parties are supposed to play
but 1 plays

- but if is equilibrium play in
then not clear from who has deviated

- resolution: punish them both with inefficient outcome a.

But what if parties can renegotiate outcome ex post ?

- not an issue when designer is third party; here parties themselves design contract
- why settle for a ?
- will renegotiate a to get something Pareto optimal
- renegotiation interferes with effective punishment: can’t punish both parties
- in Segal (1999) and Hart and Moore (1999), renegotiation is so constraining that mechanisms are useless
Risk aversion

- Pareto frontier (in utility space) is strictly concave
- so if randomize between 2 Pareto optimal points, generate point in interior (bad outcome)
- so can punish both parties after all.

Why isn’t randomization renegotiated away?

- randomization occurs only out of equilibrium
- ex ante, parties have no incentive to renegotiate (expect equilibrium
- have incentive not to renegotiate: renegotiation interferes with getting equilibrium outcome

- what about renegotiation ex post?
- create randomizing device so that as soon as a party deviates from equilibrium, randomization realized
- no time to renegotiate

Open problem:

How to provide fully satisfactory foundation for incomplete contracts?

- possible answer: bounded rationality (inability to perform dynamic programming)